Identification of critical transmission limits in injection impedance plane

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Abstract

In this paper, equations are derived that describe the mapping of critical boundaries and characteristic curves from the three dimensional PQV-surface into the two-dimensional injection impedance plane (load impedance plane for both positive and negative resistance). The expressions derived for the critical and characteristic curves in the impedance plane form the basis for a new phasor measurement based situational awareness method, which uses the results in this paper to identify critical operational boundaries in real time and to visualize the system operating conditions in an informative way. The situational awareness method will be described in a later paper, where this paper focuses on the derivations of some system characteristics in the injection (or load) impedance plane. The critical curves from the PQV-surface that are mapped into the impedance plane are the ones representing the conditions where the partial derivatives of the variables P, Q and V in respect to each other become zero. In addition to the mapping of the critical curves, some characteristic curves are mapped as well. These include the mapping of the curves of constant P, Q, V and δ from the PQV-surface into the impedance plane. All of the mapped critical and characteristic curves appear as circles in the injection impedance plane.

Highlights

► Derivations of critical boundaries for system stability in terms of injection impedance. ► Derivations of characteristic curves of constant P, Q, V and delta in injection impedance plane. ► Example of how one of the derived mappings provides an analytical load flow solution.

Introduction

As a consequence of world wide increased political focus on climate changes and thereby an increased focus on reduction of CO2 emission, it is anticipated that the share of electric power production that is based on renewable energy sources will continually increase in the coming decades. This trend is apparent in many countries where especially power production based on wind and solar energy have been rapidly increasing during the last decade.

In a power system where the share of power production by means of uncontrollable renewable energy sources (such as wind or solar energy) is gradually increasing, the power production becomes more decentralized where the production units are often relatively small and spread over a wide area in the power system. In systems where the amount of wind power production has been significantly increased, the existing transmission system is not always designed to cope with the new production patterns. In many cases, it would be desirable to strengthen existing transmission system, but due to an increasing political and public resistance against further expansion of high voltage transmission grids, future expansions could possible be limited.

Another possibility, instead of strengthening the current transmission system, would be an operation of the existing power system closer to its critical boundaries. Such system operation requires a trustworthy online monitoring of these boundaries. The formation of the power system towards increasing utilization of uncontrollable sources of renewable energy combined with generally more stressed transmission system, necessitates a research within the field of methods providing situational awareness for the power system operation in real time. A situational awareness involves both a knowledge concerning the current operating point, its distance to critical operational boundaries and knowledge of how the operating point could be controlled to increase distance from the critical boundaries.

With the introduction of phasor measurement units (PMUs) [1], [2], synchronized measurements of the system voltage and current phasors became possible. With a widespread usage of PMUs in electric power system, a time synchronized snapshots of the system conditions can be updated with a repetition rate equal the system frequency. Several approaches for optimal PMU placement to obtain full observability of the system conditions have been reported [3], [4], [5]. The high repetition frequency of PMUs measurements opens up for the development of new applications of wide-area monitoring, detection, protection and control [6], [7], [8], [9], [10], [11], [12]. Present and potential applications of phasor measurements have been documented in several surveys [13], [14], [15].

One potential application of phasor measurements is to utilize them for obtaining real time situational awareness, where the system stability boundaries are monitored in realtime. As a part of the process of developing methods that assess system stability in real time, it can be useful to express critical system boundaries in terms of measurable system quantities, such as system injection impedances (the term injection impedance is used to denote a load impedance where the resistive part can be positive or negative). Expressing the system boundaries in the injection impedance plane, real time measurements of the injection impedances can be held against the boundaries, hence providing an assessment of the system stability.

This paper describes analytical derivations for the mapping of some useful characteristics of a three dimensional PQV-surface into an injection impedance plane. The presented expressions for the characteristic boundaries form the basis of a situational awareness method that exploits results from this paper for real time stability assessment and informative visualization of observed system conditions.

Section snippets

Critical curves on a PQV-surface

The two bus system and the relevant notations for system variables used in the following derivations are provided in Fig. 1. The relationship between receiving and sending end voltage (E and V), active power P and reactive power Q can be written as:V4+V2(2(RP+XQ)-E2)+(R2+X2)(P2+Q2)=0

A solution for either P, Q or V has to be determined if a PQV-surface is to be plotted. Rearranging (1) and solving for P gives the two solutions below:P[1]=-RV2--Q2(R2+X2)2+(R2+X2)(E2-2QX)V2-X2V4R2+X2P[2]=-RV2+-Q2(R

Characteristics of the surface

As necessary background for the mapping of the critical curves on the PQV-surface into the injection impedance plane, the following relationship must be valid:WhenPV=0,thenQV=0.WhenPQ=0,thenVQ=0.WhenQP=0,thenVP=0.In the following subsections it is shown that (9), (10), (11) are valid statements.

Transformation of the critical curves into injection impedance plane

The following sections describe the mapping of the curves on the PQV-surface, that represent the set of points where the partial derivatives become zero, into the injection impedance plane.

Transformation of other characteristic curves on the PQV-surface into impedance plane

The previously derived mappings of the critical curves where the partial derivatives on the PQV-surface are zero, are useful in stability studies [10]. For the purpose of establishing meaningful visualization of a given operating condition in injection impedance plane, the mapping of curves of constant P, Q, V and δ become of interest. In the following section, the mappings of those curves are derived.

Example – analytical load flow for a two bus system

The derived expressions for the PQV-characteristics can be used to provide an analytical solution of the load flow problem for a simple two bus system. Choosing the sending end as a reference bus, the power flow solution can be analytically determined by considering the where the circles of constant P and Q intercept in the injection impedance plane.

Eq. (42) describes the curves of constant P as function of E, R, X and θ. With values of P and Q specified, θ is known and hence the load impedance

Conclusion

Analytical derivation for the mapping of critical and characteristic curves on a three dimensional PQV-surface into the injection plane were introduced in this paper. The critical curves of interest were those where the partial derivative of the variables P, Q and V in respect to each other becomes zero.

The curve satisfying ∂P/∂Q = 0 represents a maximum receivable or injectable power at a bus when the voltage magnitude in both ends is constant. This limit appears as the circle in the impedance

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